Probabilistic Analysis of Multiparameter Persistence Decompositions into Intervals

Authors Ángel Javier Alonso , Michael Kerber , Primoz Skraba



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Author Details

Ángel Javier Alonso
  • Institute of Geometry, Graz University of Technology, Austria
Michael Kerber
  • Institute of Geometry, Graz University of Technology, Austria
Primoz Skraba
  • Queen Mary University of London, United Kingdom

Acknowledgements

The authors thank Jan Jendrysiak for helpful discussions. M.K. and P.S. acknowledge the Dagstuhl Seminar 23192 “Topological Data Analysis and Applications” that initiated this collaboration.

Cite As Get BibTex

Ángel Javier Alonso, Michael Kerber, and Primoz Skraba. Probabilistic Analysis of Multiparameter Persistence Decompositions into Intervals. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 6:1-6:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.SoCG.2024.6

Abstract

Multiparameter persistence modules can be uniquely decomposed into indecomposable summands. Among these indecomposables, intervals stand out for their simplicity, making them preferable for their ease of interpretation in practical applications and their computational efficiency. Empirical observations indicate that modules that decompose into only intervals are rare. To support this observation, we show that for numerous common multiparameter constructions, such as density- or degree-Rips bifiltrations, and across a general category of point samples, the probability of the homology-induced persistence module decomposing into intervals goes to zero as the sample size goes to infinity.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Topology
Keywords
  • Topological Data Analysis
  • Multi-Parameter Persistence
  • Decomposition of persistence modules
  • Poisson point processes

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